Nonexpansive attractors with specification (Q1076386)

Let \(f: Y\to Y\) be a continuous surjection of a compact metric space. The inverse limit of f induces a compact metric space \(\bar Y\) and a homeomorphism \(\bar f\) of \(\bar Y.\) \((\bar Y,\bar f)\) is called the natural extension of f. Consider the interval \(I=[0,1]\) with the Euclidean metric and the continuous surjection \(f(x)=1-| 2x-1|\) on I. Let \((X,\sigma)\) be the natural extension of \((I,f)\). The author proves that \((X,\sigma)\) is not expansive, \((X,\sigma)\) satisfies specification, and each point of X has a neighborhood which is homeomorphic to the product of a compact interval and a Cantor set. Moreover, there exists a \(C^ 1\)- diffeomorphism g of the 3-sphere, \(S^ 3\), which has an attractor \(\Lambda\) such that \((\Lambda,g)\) is topologically conjugate to \((X,\sigma)\).